Inductive Definitions and Their Closure Ordinals

Aanderaa, Stål

doi:10.1016/s0049-237x(08)70589-5

Cited by 17 publications

(128 citation statements)

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“…By Lemma 2.2, we can find Us so that (1) Us E % (2) Ut > s, (3) Proof. Our starting point is the following well-known lemma which plays a crucial role in the paper [7].…”

Section: Wu = {X ç U\{n)x(j) -E)mentioning

confidence: 99%

Hyperarithmetically Encodable Sets

Solovay¹

1978

Transactions of the American Mathematical Society

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Abstract.We say that a set of integers, A, is hyperarithmetically (recursively) encodable, if every infinite set of integers X contains an infinite subset Y in which A is hyperarithmetical (recursive). We show that the recursively encodable sets are precisely the hyperarithmetic sets. Let a be the closure ordinal of a universal 2¡ inductive definition. Then A is hyperarithmetically encodable iff it is constructible before stage a.We also prove an effective version of the Galvin-Prikry results that open sets, and more generally Borel sets, are Ramsey, and in the case of open sets prove that our improvement is optimal.As usual, if A and B are subsets of w, A

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“…By Lemma 2.2, we can find Us so that (1) Us E % (2) Ut > s, (3) Proof. Our starting point is the following well-known lemma which plays a crucial role in the paper [7].…”

Section: Wu = {X ç U\{n)x(j) -E)mentioning

confidence: 99%

Hyperarithmetically Encodable Sets

Solovay¹

1978

Transactions of the American Mathematical Society

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“…(A) Translating reflexivity from sets of integers to ordinals via norms we can easily see that if nn ian) = least Tln (En) reflecting ordinal (as in Aczel-Richter [3]) and S^ = sup{|:J isa An ordinal}, then (usingPD) S2n + 1 =*ln + l <°ln + l for « > 0 (and also b\n = o\n <Tt\n)-For n = 1 it is known that 6} <7r} <o\; see Aanderaa [1].…”

Section: Corollarymentioning

confidence: 99%

The Theory of Countable Analytical Sets

Kechris¹

1975

Transactions of the American Mathematical Society

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ABSTRACT. The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n a largest countable II set of reals, 2). The internal structure of the sets €" is then investigated in detail, the point of departure being the fact that 1 each <2n is a set of An-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, uj-models of analysis, higherlevel analogs of the constructible universe, inductive definability, etc.It is a classical theorem of effective descriptive set theory that a 2 J thin (i.e. containing no perfect set) subset of the continuum is countable and in fact contains only A{ reals. As a consequence, among the countable 2} sets of reals there is no largest one. Solovay [41] showed that every thin 2] set contains only constructible reals and therefore (in sharp contrast with the previous case), assuming a measurable cardinal exists, there is a largest countable 1,\ set of reals, namely the set of constructible ones. In [19] Moschovakis and the author extended Solovay's theorem to all even levels of the analytical hierarchy. It was proved there that, assuming projective determinacy (PD), there is a largest countable X\n set f°r all n> 1, which we denote by C2"-The first main result we prove in this paper (see §1) is the existence of a largest countable Il2" + 1 set (which we denote by C2w + i) for all n>0, assuming PD again. For w = 0 our proof shows, in ZF + DC only, the existence of a largest thin II} set of reals C,, a fact which was also independently discovered by D. Guaspari [12] and G. E. Sacks [38]. (To complete the picture, we remark here that it was shown in [17], using PD, that no largest countable 2L+i or n2n sets exist.) Once the existence of the largest countable sets C" is established the rest of § 1 is devoted to the study of their internal structure. We show here that C" is actually a set of A¿-degrees which is iwell ordered under the usual ordering of An-degrees.

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confidence: 99%

“…Given an ordinal α, write α + for the smallest admissible ordinal greater than α. Aczel and Richter [3] showed that 1 1 = 1 1 and that a countable ordinal α is Π 1 1 -reflecting if, and only if, it is α + -stable. Afterwards, Aanderaa [1] showed that 1 1 < 1 1 . Gostanian [8] showed that 1 1 is smaller than the least α which is (α + + 1)-stable; moreover, he showed that any α which is both (α + + 1)-stable and locally countable is also Σ 1 1reflecting.…”

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The Order of Reflection

Aguilera¹

2021

J. symb. log.

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Extending Aanderaa’s classical result that $\pi ^{1}_{1} < \sigma ^{1}_{1}$ , we determine the order between any two patterns of iterated $\Sigma ^{1}_{1}$ - and $\Pi ^{1}_{1}$ -reflection on ordinals. We show that this order of linear reflection is a prewellordering of length $\omega ^{\omega }$ . This requires considering the relationship between linear and some non-linear reflection patterns, such as $\sigma \wedge \pi $ , the pattern of simultaneous $\Sigma ^{1}_{1}$ - and $\Pi ^{1}_{1}$ -reflection. The proofs involve linking the lengths of $\alpha $ -recursive wellorderings to various forms of stability and reflection properties satisfied by ordinals $\alpha $ within standard and non-standard models of set theory.

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Inductive Definitions and Their Closure Ordinals

Cited by 17 publications

References 5 publications

Hyperarithmetically Encodable Sets

Hyperarithmetically Encodable Sets

The Theory of Countable Analytical Sets

The Order of Reflection

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